Question

tan 0/(1 - cot 0)+cot 0/(1 - tan 0)
= (1 + sec 0 cosec )​

Answer 1

Tanθ/(1 - Cotθ)  + Cotθ/(1 - Tanθ)  = (1 + SecθCoscθ)

Step-by-step explanation:

Tanθ/(1 - Cotθ)  + Cotθ/(1 - Tanθ)  = (1 + SecθCoscθ)

LHS

= Tanθ/(1 - Cotθ)  + Cotθ/(1 - Tanθ)

= (Sinθ/Cosθ)/(1 - Cosθ/Sinθ)  +  (Cosθ/Sinθ)/(1 - Sinθ/Cosθ)

= Sin²θ/(Cosθ(Sinθ - Cosθ))   + Cos²θ/(Sinθ(Cosθ - Sinθ))

= Sin²θ/(Cosθ(Sinθ - Cosθ))   - Cos²θ/(Sinθ(Sinθ - Cosθ))

= (Sin³θ - Cos³θ)/(CosθSinθ(Sinθ - Cosθ))

using a³ - b³ = (a - b) (a² + b² + ab)

= (Sinθ - Cosθ)(Sin²θ + Cos²θ + SinθCosθ)/ (CosθSinθ(Sinθ - Cosθ))

= (1 + SinθCosθ)/ (CosθSinθ)

= 1/ (CosθSinθ)  + 1

= SecθCosecθ  + 1

= 1 +  SecθCosecθ

= RHS

QED

proved

Tanθ/(1 - Cotθ)  + Cotθ/(1 - Tanθ)  = (1 + SecθCoscθ)

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